# Detection, Estimation modulation theory part 1 - Vantress H.

ISBN 0-471-09517-6

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Colored noise problem

Possible approaches: Karhunen-Lo6ve expansion, prewhitening filter, generation of sufficient statistic, 289

Reversibility proof (this idea is used several times in the text), 289-290 Whitening derivation, 290-297

Define whitening filter, hw(t, u), 291 Define inverse kernel Qn(t, u) and function for correlator g(t), 292

Derive integral equations for above functions, 293297

Draw the three realizations for optimum receiver (Fig. 4.38), 293 Construction of Qn(t, u)

Interpretation as impulse minus optimum linear filter, 294-295 Series solutions, 296-297 Performance, 301-307

Expression for d2 as quadratic form and in terms of eigenvalues, 302

Optimum signal design, 302-303 Singularity, 303-305

Importance of white noise assumption and effect of removing it

Duality with known channel problem, 331-333

Assign as reading,

1. Estimation (4.3.5)

2. Solution to integral equations (4.3.6)

3. Sensitivity (4.3.7)

4. Known linear channels (4.3.8)

Problem Set 7 (continued)

4. 4.3.4 7. 4.3.12

5. 4.3.7 8. 4.3.21

6. 4.3.8

652 Appendix: A Typical Course Outline

4.4.1

4.4.2

45-4.7

Lecture 16 pp. 333-377

Signals with unwanted parameters, 333-366

Example of random phase problem to motivate the model, 336

Construction of LRT by integrating out unwanted parameters, 334

Models for unwanted parameters, 334

Random phase, 335-348

Formulate bandpass model, 335-336 Go to A(r(t)) by inspection, define quadrature sufficient statistics Lc and Ls, 337 Introduce phase density (364), motivate by brief phaselock loop discussion, 337 Obtain LRT, discuss properties of In I0(x), point out it can be eliminated here but will be needed in diversity problems, 338-341 Compute ROC for uniform phase case, 344 Introduce Marcum’s Q function Discuss extension to binary and M-ary case, signal selection in partly coherent channels

Random amplitude and phase, 349-366

Motivate Rayleigh channel model, piecewise constant approximation, possibility of continuous measurement, 349-352

Formulate in terms of quadrature components, 352 Solve general Gaussian problem, 352-353 Interpret as filter-squarer receiver and estimator-correlator receiver, 354 Apply Gaussian result to Rayleigh channel, 355 Point out that performance was already computed in Chapter 2

Discuss Rician channel, modifications necessary to obtain receiver, relation to partially-coherent channel in Section 4.4.1, 360-364

Assign Section 4.6 to be read before next lecture; Sections 4.5 and 4.7 may be read later, 366-377

Lecture 16 653

Lecture 16 (continued)

Problem Assignment 8

1. 4.4.3 5. 4.4.29

2. 4.4.5 6. 4.4.42

3. 4.4.13 7. 4.6.6

4. 4.4.27

654 Appendix: A Typical Course Outline

Lecture 17 pp. 370-460

4.6 Multiple-parameter estimation, 370-374

Set up a model and derive MAP equations for the

colored noise case, 374

The examples can be left as a reading assignment but

the MAP equations are needed for the next topic

Chapter 5 Continuous waveform estimation, 423-460

5.1, 5.2 Model of problem, typical continuous systems such

as AM, PM, and FM; other problems such as

channel estimation; linear and nonlinear modula

tion, 423-426

Restriction to no-memory modulation, 427

Definition of am&v(r{t)) in terms of an orthogonal

expansion, complete equivalence to multiple para

meter problem, 429-430

Derivation of MAP equations (31-33), 427-431

Block diagram interpretation, 432-433

Conditions for an efficient estimate to exist, 439

5.3-5.6 Assign remainder of Chapter 5 for reading, 433-460

Lecture 18

Lecture 18 655

pp. 467-481

Chapter 6 6.1

Linear modulation

Model for linear problem, equations for MAP interval estimation, 467-468

Property 1: MAP estimate can be obtained by

using linear processor; derive integral equation, 468 Property 2: MAP and MMSE estimates coincide

for linear modulation because efficient estimate exists, 470 Formulation of linear point estimation problem, 470 Gaussian assumption, 471 Structured approach, linear processors

Property 3: Derivation of optimum linear pro-

cessor, 472

Property 4: Derivation of error expression [em-

phasize (27)], 473 Property 5: Summary of information needed, 474 Property 6: Optimum error and received wave-

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